Almost fixed-point-free automorphisms of prime order

Bertram Wehrfritz

Open Mathematics (2011)

  • Volume: 9, Issue: 3, page 616-626
  • ISSN: 2391-5455

Abstract

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Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.

How to cite

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Bertram Wehrfritz. "Almost fixed-point-free automorphisms of prime order." Open Mathematics 9.3 (2011): 616-626. <http://eudml.org/doc/269477>.

@article{BertramWehrfritz2011,
abstract = {Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.},
author = {Bertram Wehrfritz},
journal = {Open Mathematics},
keywords = {Soluble group; Groups of finite Hirsch number; Groups of finite rank; Fixed points of automorphisms of prime order; soluble groups; groups of finite Hirsch number; groups of finite rank; automorphisms of prime order; fixed-point-free automorphisms; characteristic subgroups of finite index},
language = {eng},
number = {3},
pages = {616-626},
title = {Almost fixed-point-free automorphisms of prime order},
url = {http://eudml.org/doc/269477},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Bertram Wehrfritz
TI - Almost fixed-point-free automorphisms of prime order
JO - Open Mathematics
PY - 2011
VL - 9
IS - 3
SP - 616
EP - 626
AB - Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.
LA - eng
KW - Soluble group; Groups of finite Hirsch number; Groups of finite rank; Fixed points of automorphisms of prime order; soluble groups; groups of finite Hirsch number; groups of finite rank; automorphisms of prime order; fixed-point-free automorphisms; characteristic subgroups of finite index
UR - http://eudml.org/doc/269477
ER -

References

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  9. [9] Mal’tsev A.I., On certain classes of infinite soluble groups, Mat. Sb., 1951, 28(3), 567–588 (in Russian) 
  10. [10] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups. I–II, Ergeb. Math. Grenzgeb., 62–63, Springer, Berlin-Heidelberg-New York, 1972 
  11. [11] Wehrfritz B.A.F., Infinite Linear Groups, Ergeb. Math. Grenzgeb., 76, Springer, Berlin-Heidelberg-New York, 1973 
  12. [12] Wehrfritz B.A.F., Groupand Ring Theoretic Properties of Polycyclic Groups, Algebr. Appl., 10, Springer, Dordrecht, 2009 http://dx.doi.org/10.1007/978-1-84882-941-1 
  13. [13] Wehrfritz B.A.F., Almost fixed-point-free automorphisms of soluble groups, J. Pure Appl. Algebra, 2011, 215(5), 1112–1115 http://dx.doi.org/10.1016/j.jpaa.2010.07.017 Zbl1215.20030
  14. [14] Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press) 

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